Box 14 of my W2 has lines for NYDBL and NYPFL. When entering these into Turbotax, what do I put for Box 14-Category?

These taxes (Disability Benefits Law and Paid Family Leave) are federally deductible (though with tax reform changes it will be harder to see a tax benefit).  You should choose "Other mandatory deductible state or local tax not on above list" - see screenshot below.

Thank you.  I also have a line that says BONUS.  What should I choose for that one under Box 14-Category?

Box 14 can be used to report pretty much anything that the employer chooses and I would assume that your employer is just reporting that there is a bonus payment included in your wages.  You would just want to choose "Other - not on above list" in that case - note that this is a separate Other selection than the one above as this would not be a deductible tax.  But, you may want to confirm what that line actually represents with your employer.

That is NOT an option? NOw what